Planar heating element

Abstract

This invention discloses a planar heating element that achieves a good balance between uniform heating of the substrate and suppression of electromagnetic wave diffraction. [Solution] The planar heating element 1 includes a transparent substrate 10 and an electrical conductor 20 on the substrate 10. The electrical conductor 20 has a network structure. The network structure has the shape of a two-dimensional array of Jordan curves 30 that are congruent with respect to translation. Each Jordan curve 30 is composed of a finite-length reference curve 31 with endpoints, curves 32 and 33 obtained by a mirror transformation of the reference curve 31 with respect to each of the two orthogonal axes, and a curve 34 obtained by an inversion transformation of the reference curve 31. The reference curve 31 is represented as the graph of a strictly monotonically increasing continuous function f(x):[a,b]→R defined on the closed interval [a,b] which is a subset of the set R of all real numbers.

Description

[Technical Field]

[0001] This disclosure relates to a transparent planar heating element. [Background technology]

[0002] Regarding transparent planar heating elements, a planar heating element comprising a transparent substrate and an electrical conductor on this substrate is known (see Patent Documents 1, 2, 3, and 4).

[0003] The reason why planar heating elements include a transparent substrate is that electromagnetic waves such as visible light and infrared rays are expected to pass through the heating element. Such planar heating elements are used, for example, as heaters for window glass and heaters for distance measuring sensors mounted on automobiles. For this reason, the electrical conductors on the substrate generally have a wiring pattern that does not obstruct electromagnetic waves as much as possible and can heat the substrate uniformly. [Prior art documents] [Patent Documents]

[0004] [Patent Document 1] Japanese Patent No. 6706299 [Patent Document 2] International Publication Number WO2023 / 120024 [Patent Document 3] U.S. Patent No. 9967922 [Patent Document 4] Japanese Patent Publication No. 2010-003667 [Overview of the project] [Problems that the invention aims to solve]

[0005] From the viewpoint of uniform heating of the substrate, it is desirable for the wiring pattern of electrical conductors on a transparent substrate to have a regular geometric shape. Furthermore, from the viewpoint of reducing electrical resistance, it is desirable for the regular geometric shape to have many straight lines parallel or approximately parallel to the direction in which the voltage is applied to the electrical conductor. However, a regular geometric shape with many straight lines induces the diffraction phenomenon of electromagnetic waves. To suppress the diffraction phenomenon of electromagnetic waves, it is desirable for the wiring pattern of electrical conductors not to have a regular geometric shape, or even if it does have a regular geometric shape, it is desirable that the regular geometric shape does not have straight lines. Thus, uniform heating of the substrate and suppression of the diffraction phenomenon of electromagnetic waves are in a trade-off relationship with each other.

[0006] This specification discloses a planar heating element that achieves a good balance between uniform heating of a substrate and suppression of electromagnetic wave diffraction. [Means for solving the problem]

[0007] The technical matters described herein are provided not to explicitly or implicitly limit the invention described in the claims, nor to enable persons other than those who benefit from the invention (e.g., the applicant and the rights holder) to limit the invention described in the claims, but simply to facilitate understanding of the essential points of the invention. An overview of the invention from other perspectives can be understood, for example, from the claims of this patent application as of the filing date. The disclosed planar heating element comprises a transparent substrate and an electrical conductor on the substrate. The electrical conductor has a network structure. The network structure has the shape of a two-dimensional array of Jordan curves that are congruent with respect to translation. Each Jordan curve is composed of a finite-length reference curve with endpoints, a curve obtained by a reflection transformation of the reference curve with respect to each of the two orthogonal axes, and a curve obtained by an inversion transformation of the reference curve. The reference curve is represented as the graph of a strictly monotonically increasing continuous function f(x):[a,b]→R defined on the closed interval [a,b], which is a subset of the set of all real numbers R. 1) The equation 3(f(b)-f(a)) / 2≦ba≦6(f(b)-f(a)) holds true. 2) The function f(x) is first differentiable on the open interval (a,b), 3)

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[0008] The planar heating element of this disclosure can achieve a good balance between uniform heating of the substrate and suppression of electromagnetic wave diffraction. [Brief explanation of the drawing]

[0009] [Figure 1] Structure of a first example of a planar heating element according to the first embodiment. (a) Plan view. (b) Jordan curve. [Figure 2] A diagram illustrating the reference curve. [Figure 3] A diagram illustrating wavy filaments. [Figure 4] Structure of a second example of the planar heating element of the first embodiment. (a) Plan view. (b) Jordan curve. [Figure 5] Structure of a third example of a planar heating element according to the first embodiment. (a) Plan view. (b) Jordan curve. [Figure 6] Structure of a planar heating element according to the second embodiment. [Figure 7]Structure of a planar heating element according to the third embodiment (first example). [Figure 8] Structure of a planar heating element according to the third embodiment (second example). [Figure 9] Structure of a planar heating element according to the third embodiment (third example). [Modes for carrying out the invention]

[0010] The planar heating element disclosed will be described with reference to the drawings. Note that the drawings are for understanding the embodiments, and the scale of the illustrated components differs from the actual scale. Furthermore, in each drawing, reference numerals are used only for some of the two or more identical components to improve readability.

[0011] <First Embodiment> Figure 1 shows a first example of the planar heating element 1 of the first embodiment, Figure 4 shows a second example, and Figure 5 shows a third example. The planar heating element 1 of the first embodiment includes a transparent substrate 10 and an electrical conductor 20 on the substrate 10. The substrate 10 is, for example, a flat plate or film-shaped transparent resin cured material, or a flat plate-shaped glass. The electrical conductor 20 is formed on the substrate 10 by printing (gravure offset printing, screen printing, etc.), photolithography, etc. The electrical conductor 20 has a mesh structure. The mesh structure has the shape of a two-dimensional array of many Jordan curves (Jordan curves are also called simple closed curves) 30 that are congruent with respect to translation. Any two adjacent Jordan curves 30 among the many Jordan curves 30 are linked to each other in a first direction or a second direction. The first direction and the second direction are orthogonal to each other. As will be described later, the mesh structure has one or two types of Jordan curves.

[0012] Each of the numerous Jordan curves 30 is composed of only four curves that neither overlap nor intersect with each other. One of the four curves, curve 31, is a finite-length curve with two endpoints (hereinafter referred to as the reference curve), one of the remaining three curves, curve 32, is obtained by the reflection of the reference curve with respect to the first direction, one of the remaining two curves, curve 33, is obtained by the reflection of the reference curve with respect to the second direction, and the remaining curve 34 is obtained by the inversion of the reference curve. Thus, each Jordan curve 30 has a shape that is symmetrical with respect to each of the two axes of symmetry (i.e., the axis parallel to the first direction and the axis parallel to the second direction) (see Figures 1(b), 4(b), and 5(b)).

[0013] The reference curve (i.e., curve 31) is represented as the graph of a strictly monotonically increasing and continuous function f(x):[a,b]→R defined on the closed interval [a,b] (where a≠b), which is a subset of the set of all real numbers R (see Figure 2). This function f(x) satisfies the following conditions 1) to 6).

[0014] Condition 1) The equation 3(f(b)-f(a)) / 2≦ba≦6(f(b)-f(a)) holds true.

[0015] Condition 2) The function f(x) is first-order differentiable on the open interval (a,b).

[0016] Condition 3)

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[0017] Condition 4) The graph of the function f(x) has one inflection point (x poi ,f(x poi )) has (a <x poi <b)。

[0018] Condition 5)

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[0019] Condition 6)

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[0020] Thus, each Jordan curve 30 is, 1) It does not have a straight segment, and the curved segment that can be approximated as a straight line is sufficiently short, 2) It is symmetric with respect to each of the two axes of symmetry (i.e., the axis parallel to the first direction and the axis parallel to the second direction), and 3) The Jordan curve 30 is sharply pointed toward the endpoint in the region near each of the two endpoints in the longitudinal direction (the second direction in the example shown in the figure), and 4) In the short direction of the Jordan curve 30 (the first direction in the example shown in the figure), the curve is obtusely gradual toward the endpoint in the neighborhood region of each of the endpoints. The electrical conductor 20 has a specific shape. Therefore, the network structure of the electrical conductor 20 has a regular geometric shape without linear components, and furthermore, it has wavy filaments 20a that extend in the longitudinal direction of the Jordan curve 30 (the second direction in the example shown in the figure) and have a relatively small amplitude and change gently in a wave-like manner in the short direction of the Jordan curve 30 (the first direction in the example shown in the figure). As shown in Figure 3, the wavy filaments 20a are filaments that have the shape of a curve in which curves 31 and 32 are alternately connected in the second direction (or a curve in which curves 33 and 34 are alternately connected in the second direction) (in Figure 3, for clarity, the two wavy filaments 20a are shown as thick lines). Because the electrical conductor 20 has such a network structure, the planar heating element 1 can achieve a good balance between uniform heating of the substrate 10 and suppression of electromagnetic wave diffraction.

[0021] [OP1] From the viewpoint of uniform heating of the substrate 10, the x-coordinate of the inflection point x poi Regarding

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[0022] [OP2] From the viewpoint of uniform heating of the substrate 10, with respect to the y-coordinate of the inflection point,

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[0023] [OP3] From the perspective of reducing electrical resistance,

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[0024] [OP4] The reference curve is, for example, a part of a sigmoid curve, or a high-order spline curve, or a B-spline curve. Examples of sigmoid curves include the logistic function (this example includes sigmoid functions and hyperbolic tangent functions), the cumulative distribution function of the normal distribution, the Gompertz function, the inverse tangent function, the Gooderman function, sin(arctan(αx)), αx / (β + γ|x|) (α, β, γ ∈ R). Since the sigmoid curve is generally defined on (-∞, ∞), a part of the sigmoid curve is a sigmoid curve on the closed interval [a, b]. In this case, a < 0 and 0 < b. By appropriately setting the parameters defining the sigmoid curve, a reference curve satisfying the above conditions 1) to 6) can be accurately obtained. Examples of B-spline curves include Bézier curves. When the reference curve is a high-order spline curve or a B-spline curve, by appropriately setting four control points, various reference curves satisfying the above conditions 1) to 6) can be easily obtained.

[0025] [OP5] The reference curve may have a point-symmetric shape. When the reference curve has a point-symmetric shape, the network structure has one type of Jordan curve 30 (see Fig. 1(a)), but when the reference curve does not have a point-symmetric shape, as a result of a two-dimensional array of one type of Jordan curve 30 that are congruent to each other with respect to translation, the network structure has two types of Jordan curves 30 (see Figs. 4(a) and 5(a)).

[0026] [OP6] The function f(x) is

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[0027] [OP7] Regarding the function f(x) described in [OP6], preferably,

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[0028] [OP8] From the viewpoint of reducing electrical resistance, the x-axis direction of the function f(x) is parallel to the second direction, and it is desirable that a voltage be applied to the electrical conductor 20 in the second direction.

[0029] <Second Embodiment> An example of the planar heating element 2 of the second embodiment is shown in FIG. 6. The planar heating element 2 of the second embodiment has a feature that in the short-side direction of the Jordan curve (the first direction in the example shown in FIG. 6), the width of the Jordan curve in a region (hereinafter referred to as the far region) away from the center of the planar heating element 2 is wider than the width of the Jordan curve in the central region of the planar heating element 2. Generally, since the electromagnetic wave source or the electromagnetic wave detector is arranged opposite to the center of the planar heating element 2, according to the planar heating element 2 of the second embodiment, the viewing angle of the Jordan curve in the far region as seen from the electromagnetic wave source or the electromagnetic wave detector is not greatly different from the viewing angle of the Jordan curve in the central region. Therefore, the planar heating element 2 is beneficial from the viewpoint of suppressing the diffraction phenomenon of electromagnetic waves. The specific configuration of the planar heating element 2 is as follows. <00​​​​​​​​​The elements are arranged in the first direction in an order that follows the order relation of the elements of the set {x∈N: 1≦x≦M}, where N is the set of all positive integers.

[0032] For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, there are M rectangular regions G1, G2, ..., G M The mth rectangular region G m A portion of the network structure contained within is a Jordan curve JC that is congruent to each other with respect to translation. m It has the shape of a one-dimensional or two-dimensional array. In the case of a two-dimensional array, it is a Jordan curve JC m Any two adjacent Jordan curves JC m They are linked to each other in either a first or second direction. The second direction is orthogonal to the first direction. For a one-dimensional array, the Jordan curve JC m Any two adjacent Jordan curves JC m They are linked to each other in a second direction.

[0033] For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the Jordan curve JC m Each of these is composed of only four curves that neither overlap nor intersect with each other. One of the four curves is a finite-length curve with endpoints (i.e., a reference curve) RC m And one of the remaining three curves is the reference curve RC in the first direction. m This is the curve obtained by reflection, and one of the remaining two curves is the reference curve RC in the second direction. m The curve obtained by reflection is the reference curve RC. m This is the curve obtained by inverting the Jordan curve JC. m It has a shape that is symmetrical with respect to each of the two axes of symmetry (i.e., the axis parallel to the first direction and the axis parallel to the second direction).

[0034] For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the reference curve RC mf is a strictly monotonically increasing and continuous function defined on the closed interval [a,b] (where a≠b), which is a subset of the set of all real numbers R. m (x):[a,b]→R is represented as a graph. This function f m (x) satisfies the following conditions 1) to 6).

[0035] Condition 1) 3(f m (b)-f m (a) / 2≦ba≦6(f m (b)-f m (a)) is true.

[0036] Condition 2) function f m (x) is first differentiable on the open interval (a,b).

[0037] Condition 3)

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[0038] Condition 4) function f m The graph of (x) has one inflection point (x poi ,f m (x poi )) has (a <x poi <b)。

[0039] Condition 5)

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[0040] Condition 6)

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[0041] moreover, For (p,q)∈{x∈N:1≦x≦M}×{x∈N:1≦x≦M} that satisfy O1) p≠q, p+q=M+1, f p (b)-f p (a) = f q (b)-f q (a) This is established, O2) For any r,s ∈ {x ∈ N: 1 ≤ x ≤ (M+1) / 2}, r <s⇒f r (b)-f r (a) > f s (b)-f s (a) This is true.

[0042] As a simple example, for each m ∈ {x ∈ N: 1 ≤ x ≤ M} f m (x=γ m ×f1(x) holds true. γ m ∈R is the scale parameter. 1=γ1=γ M >γ2=γ M-1 >...>γ (M-1) / 2 =γ (M+3) / 2 >γ (M+1) / 2 That is the case.

[0043] For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the m-th rectangular region G m The Jordan curve JC is included in this. m Jordan curve JC in the shorter direction (the first direction in the example shown in Figure 6) m The number T m There is no limitation. For any r, s ∈ {x ∈ N: 1 ≤ x ≤ M}, r ≠ s, T r =T s But that's fine too. Alternatively, for (p,q)∈{x∈N:1≦x≦M}×{x∈N:1≦x≦M} that satisfy p≠q,p+q=M+1, Tp =T q And for any r,s∈{x∈N:1≦x≦M / 2}, r <s⇒T r <T s The following may also be true (see Figure 6).

[0044] In the second embodiment, function f m With respect to (x)(m∈{x∈N:1≦x≦M}), any of [OP1] to [OP8] described in the first embodiment, or two or more combinations thereof that are compatible with each other, may be true.

[0045] <Third Embodiment> Figure 7 shows the first example of the planar heating element 3 of the third embodiment, Figure 8 shows the second example, and Figure 9 shows the third example. Generally, a second electrical conductor connected to a first electrical conductor having a mesh structure applies a voltage to the first electrical conductor. Due to the resistance of the second electrical conductor, a larger voltage drop occurs within the second electrical conductor the further it is from the power supply point on the second electrical conductor. In other words, the magnitude of the current passing through a part of the mesh structure in a region far from the power supply point is different from the magnitude of the current passing through a part of the mesh structure in a region close to the power supply point. From the viewpoint of uniform heating of the substrate in such a case, the planar heating element 3 of the third embodiment has the characteristic that, in the short-side direction of the Jordan curve, the width of the Jordan curve in the region far from the power supply point is narrower than the width of the Jordan curve in the region close to the power supply point. The specific configuration of the planar heating element 3 is as follows.

[0046] The planar heating element 3 of the third embodiment includes a transparent substrate 10, a first electrical conductor 40 on the substrate 10, and two second electrical conductors 50, each having one or two predetermined power supply points 51. The substrate 10 is, for example, a flat plate or film-shaped transparent resin cured material, or a flat plate of glass. The first electrical conductor 40 and the second electrical conductors 50 are formed on the substrate 10 by printing (gravure offset printing, screen printing, etc.), photolithography, etc. The second electrical conductors 50 may also be busbars.

[0047] The first electrical conductor 40 has M rectangular regions G1, G2, ..., G in the first direction. M It has a network structure divided into the following: M is a predetermined integer greater than or equal to 3 (in the example shown in Figure 7, M=7; in the example shown in Figure 8, M=5; and in the example shown in Figure 9, M=9). There are M rectangular regions G1, G2, ..., G M The elements are arranged in the first direction in an order that follows the order relation of the elements of the set {x∈N: 1≦x≦M}, where N is the set of all positive integers.

[0048] For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, there are M rectangular regions G1, G2, ..., G M The mth rectangular region G m A portion of the network structure contained within is a Jordan curve JC that is congruent to each other with respect to translation. m It has the shape of a one-dimensional or two-dimensional array. In the case of a two-dimensional array, it is a Jordan curve JC m Any two adjacent Jordan curves JC m They are linked to each other in either a first or second direction. The second direction is orthogonal to the first direction. For a one-dimensional array, the Jordan curve JC m Any two adjacent Jordan curves JC m They are linked to each other in a second direction.

[0049] For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the Jordan curve JC m Each of these is composed of only four curves that neither overlap nor intersect with each other. One of the four curves is a finite-length curve with endpoints (i.e., a reference curve) RC m And one of the remaining three curves is the reference curve RC in the first direction. m This is the curve obtained by reflection, and one of the remaining two curves is the reference curve RC in the second direction. m The curve obtained by reflection is the reference curve RC. m This is the curve obtained by inverting the Jordan curve JC. mhas a shape that is line-symmetric with respect to each of two symmetry axes (i.e., an axis parallel to the first direction and an axis parallel to the second direction).

[0050] For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the reference curve RC m is represented as the graph of a strictly monotone increasing and continuous function f m (x): [a, b] → R defined on a closed interval [a, b] (where a ≠ b), which is a subset of the set R of all real numbers. This function f m (x) satisfies the following conditions 1) to 6).

[0051] Condition 1) 3(f m (b) - f m (a)) / 2 ≤ b - a ≤ 6(f m (b) - f m (a)) holds.

[0052] Condition 2) The function f m (x) is differentiable in the open interval (a, b).

[0053] Condition 3)

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[0054] Condition 4) The graph of the function f m (x) has one inflection point (x poi , f m (x poi )) (a < x poi<b).

[0055] Condition 5) [Number] is satisfied.

[0056] Condition 6) [Number] is satisfied.

[0057] Furthermore, when one power supply point 51 is predetermined for each of the two second electric conductors 50: The power supply point 51 of one of the two second electric conductors 50 and the power supply point 51 of the other second electric conductor 50 are located on a straight line passing through the K-th rectangular region G M among M rectangular regions G1, G2,..., G K parallel to the second direction.

[0058] (i - 1) When 1 < K < M (see Fig. 7), For any r, s ∈ {x ∈ N: 1 ≤ x ≤ K}, r < s ⇒ f r (b) - f r (a) < f s (b) - f s (a) is satisfied, For any r, s ∈ {x ∈ N: K ≤ x ≤ M}, <​​​​​​​​​​​​​​​​​​​​​​​​​​This is true.

[0059] (ii) When two power supply points 51 are predetermined in each of the two second electrical conductors 50 (see Figure 9): Two power supply points 51 are located at both ends of one of the two second electrical conductors 50, and two power supply points 51 are located at both ends of the other second electrical conductor 50. M is an odd number. For any r,s ∈ {x ∈ N: 1 ≤ x ≤ (M+1) / 2}, r <s⇒f r (b)-f r (a) > f s (b)-f s (a) This is established, For any r,s ∈ {x ∈ N: (M+1) / 2 ≤ x ≤ M}, r <s⇒f r (b)-f r (a) <f s (b)-f s (a) This is true.

[0060] As a simple example, for each m ∈ {x ∈ N: 1 ≤ x ≤ M} f m (x=γ m ×f1(x) holds true. γ m ∈R is the scale parameter. In the case of (i-1), 1 = γ1 < γ2 < ... < γ K-1 <γ K >γ K+1 >...>γ M That is the case. In the case of (i-2), 1 = γ1 < γ2 < ... < γ K-1 <γ K That is the case. (ii) In this case, 1 = γ1 > γ2 > ... > γ K-1 >γ K <γ K+1 <...<γ M That is the case.

[0061] In the third embodiment, function f mWith respect to (x)(m∈{x∈N:1≦x≦M}), any of [OP1] to [OP8] described in the first embodiment, or two or more combinations thereof that are compatible with each other, may be true.

[0062] <Addendum 1> The technical features disclosed in the various embodiments and their variations described above are not necessarily mutually exclusive. To the extent that they do not contradict each other from a technical standpoint, the technical features of one embodiment or its variation may be applied to the technical features of another embodiment or its variation.

[0063] The claims set forth in the claims of this application at the time of filing do not necessarily claim all inventions disclosed in this specification. In this regard, the applicant of this application should not be understood or interpreted as having waived the right to obtain a patent for inventions not claimed at the time of filing this application. To the extent permitted by the laws or treaties of the country or region that receives this application, the applicant of this application reserves the right to obtain a patent for inventions not claimed in this application, the right to file a divisional application for such inventions, the right to claim such inventions by amendment, and all other rights. However, this shall not apply if the applicant of this application expresses an explicit and definitive contrary intention.

[0064] An example of a summary of this disclosure from a different perspective is as follows:

[0065] A planar heating element based on the first perspective is: A transparent substrate, The electrical conductor on the substrate and Includes, The aforementioned electrical conductor has a network structure, The network structure has the shape of a two-dimensional array of Jordan curves that are congruent with respect to translation, wherein any two adjacent Jordan curves are linked to each other in a first direction or a second direction, and the first and second directions are orthogonal to each other. Each of the Jordan curves is composed of only four curves that neither overlap nor intersect with each other, wherein one of the four curves is a finite-length reference curve with endpoints, one of the remaining three curves is obtained by the reflection of the reference curve in the first direction, one of the remaining two curves is obtained by the reflection of the reference curve in the second direction, and the remaining one curve is obtained by the inversion of the reference curve. The aforementioned reference curve is represented as the graph of a strictly monotonically increasing continuous function f(x):[a,b]→R defined on the closed interval [a,b], which is a subset of the set of all real numbers R, where, 1) The equation 3(f(b)-f(a)) / 2≦ba≦6(f(b)-f(a)) holds true. 2) The function f(x) is first differentiable on the open interval (a,b), 3)

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[0066] A planar heating element based on the second perspective is a planar heating element based on the first perspective, (2a+b) / 3≦x poi The equation ≤ (a+2b) / 3 holds true.

[0067] A planar heating element based on the third viewpoint is a planar heating element based on the first viewpoint or the second viewpoint, (2f(a)+f(b)) / 3≦f(x poi The equation )≦(f(a)+2f(b)) / 3 holds true.

[0068] A planar heating element based on the fourth viewpoint is a planar heating element based on either the first viewpoint or the third viewpoint, f'(x poi The condition )≦tan(π / 3) holds true.

[0069] A planar heating element based on the fifth viewpoint is a planar heating element based on any of the first viewpoint or the fourth viewpoint, The aforementioned reference curve is a part of a sigmoid curve, or a higher-order spline curve, or a B-spline curve. A planar heating element characterized by the following features.

[0070] A planar heating element based on the sixth viewpoint is a planar heating element based on any of the first viewpoint or the fifth viewpoint, The aforementioned reference curve is point-symmetric.

[0071] A planar heating element based on the seventh viewpoint is a planar heating element based on any of the first viewpoint or the fourth viewpoint, The above function f(x) is,

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[0072] A planar heating element based on the eighth viewpoint is a planar heating element based on either the first viewpoint or the seventh viewpoint, The x-axis direction of the function f(x) is parallel to the second direction, A voltage is applied to the electrical conductor in the second direction.

[0073] A planar heating element based on the ninth perspective is A transparent substrate, The electrical conductor on the substrate and Includes, The electrical conductor has a network structure divided into M rectangular regions in a first direction, where M is a predetermined odd number of 3 or more. The M rectangular regions are arranged in the first direction in an order according to the order relation of the elements of the set {x∈N:1≦x≦M}, where N is the set of all positive integers. For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, a portion of the network structure contained within the m-th rectangular region out of the M rectangular regions is a Jordan curve JC that is congruent to each other with respect to translation. m It has the shape of a one-dimensional or two-dimensional array, however, in the case of a two-dimensional array, the Jordan curve JC m Any two adjacent Jordan curves JC m The elements are linked to each other in the first or second direction, the second direction being orthogonal to the first direction, and in the case of a one-dimensional array, the Jordan curve JC m Any two adjacent Jordan curves JC m They are linked to each other in the direction of the second tier, For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the Jordan curve JC m Each of these is composed of only four curves that neither overlap nor intersect with each other, except that one of the four curves is a finite-length reference curve RC with endpoints. m And one of the remaining three curves is the reference curve RC in the first direction. m The curve obtained by reflection is the curve of the second direction, and one of the remaining two curves is the reference curve RC in the second direction. m The curve obtained by reflection is the reference curve RC. m This is a curve obtained by inverting the curve, For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the reference curve RC mf is a strictly monotonically increasing continuous function defined on the closed interval [a,b], which is a subset of the set of all real numbers R. m (x):[a,b]→R is represented as a graph, where, 1)3(f m (b)-f m (a) / 2≦ba≦6(f m (b)-f m (a)) is true, 2) The function f m (x) is first differentiable on the open interval (a,b), 3)

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[0074] A planar heating element based on the tenth perspective is A transparent substrate, The first electrical conductor on the substrate, Two second electrical conductors, each with one or two predetermined power supply points, Includes, The first electrical conductor has a network structure divided into M rectangular regions in a first direction, where M is a predetermined integer of 3 or more. The M rectangular regions are arranged in the first direction in an order according to the order relation of the elements of the set {x∈N:1≦x≦M}, where N is the set of all positive integers. For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, a portion of the network structure contained within the m-th rectangular region out of the M rectangular regions is a Jordan curve JC that is congruent to each other with respect to translation. m It has the shape of a one-dimensional or two-dimensional array, however, in the case of a two-dimensional array, the Jordan curve JC m Any two adjacent Jordan curves JC m The elements are linked to each other in the first or second direction, the second direction being orthogonal to the first direction, and in the case of a one-dimensional array, the Jordan curve JC m Any two adjacent Jordan curves JC m They are linked to each other in the direction of the second tier, For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the Jordan curve JC m Each of these is composed of only four curves that neither overlap nor intersect with each other, except that one of the four curves is a finite-length reference curve RC with endpoints. m And one of the remaining three curves is the reference curve RC in the first direction. m The curve obtained by reflection is the curve of the second direction, and one of the remaining two curves is the reference curve RC in the second direction. m The curve obtained by reflection is the reference curve RC. mIt is a curve obtained by inversion, For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the reference curve RC m is represented as the graph of a strictly increasing continuous function f m (x): [a, b] → R defined on a closed interval [a, b] which is a subset of the set R of all real numbers, provided that 1) 3(f m (b) - f m (a)) / 2 ≤ b - a ≤ 6(f m (b) - f m (a)) holds, 2) The function f m (x) is differentiable in the open interval (a, b), 3)

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[0075] <Addendum 2> While the present invention has been described with reference to exemplary embodiments, those skilled in the art will understand that various modifications can be made and elements can be replaced with equivalents without departing from the scope of the invention. Furthermore, many modifications can be made to adapt a particular system, device, or component thereof to the teachings of the invention without departing from the essential scope of the invention. Accordingly, the present invention is not limited to the specific embodiments disclosed for the purpose of carrying out the invention, but includes all embodiments contained in the appended claims.

[0076] Furthermore, the use of terms such as “first,” “second,” etc., when used herein and / or in the appended claims, does not indicate order or importance, but rather the terms such as “first,” “second,” etc., are used to distinguish elements. The terms used herein are for the purpose of describing embodiments and are not intended in any way to limit the invention. The terms “including” and their variations, when used herein and / or in the appended claims, indicate the existence of the mentioned features, steps, operations, elements, and / or components, but do not exclude the existence or addition of one or more other features, steps, operations, elements, components, and / or groups thereof. The terms “and / or” include, if any, one or any combination of the listed elements relating thereto. In the claims and specification, unless otherwise specifically stated, “connected,” “joined,” “joined,” “linked,” or their synonyms, and all their forms, do not necessarily negate the existence of one or more intermediate elements between two that are, for example, “connected” or “joined” or “linked” to one another. In the claims and specification, the term “arbitrary” should be understood as having the same meaning as the universal quantifier ∀, if any, unless otherwise specified. For example, the expression “for any X” is the same as “for all X” or “for each X.” Expressions such as “at least one of A, B, and C” (for example in English “at least one of A, B and C”, “at least one of A, B or C”, “at least one of A, B and / or C”) should be understood as having the same meaning as the power set 2 of the set S which contains all the listed elements, if any, unless otherwise specified. S This means arbitrarily selecting an element from the set P obtained by removing the empty set φ from the set. In this example, S={A,B,C},2 S={φ,{A},{B},{C},{A,B},{A,C},{B,C},{A,B,C}},P={{A},{B},{C},{A,B},{A,C},{B,C},{A,B,C}}, and this example means that one element (for example, {A,C}) can be arbitrarily selected from the set P.

[0077] Unless otherwise specified, all terms used herein (including technical and scientific terms) have the same meaning as those generally understood by those skilled in the art to which the present invention pertains. Furthermore, terms such as those defined in commonly used dictionaries should be construed to have the meaning consistent with their meanings in the relevant art and in the context of this disclosure, and should not be construed ideally or excessively formally unless expressly defined.

[0078] It will be understood that many techniques and steps are disclosed in the description of this invention. Each of these has its own advantages, and each can be used in combination with one or more, or possibly all, of the other disclosed techniques. Therefore, to avoid complexity, this specification refrains from describing every possible combination of individual techniques or steps. Nevertheless, the specification and claims should be read with the understanding that such combinations are entirely within the scope of the invention and claims.

[0079] In the following claims, all corresponding structures, materials, actions, and equivalents of functional elements combined with means or steps are intended to include structures, materials, or actions for performing a function in combination with other elements, if any.

[0080] While embodiments of the present invention have been described above, the present invention is not limited to these embodiments. Various modifications and variations are permitted without departing from the spirit of the invention. The selected and described embodiments are for illustrating the principles of the present invention and its practical applications. The present invention can be used in various embodiments with various modifications and variations, and the various modifications and variations will be determined according to the expected use. All such modifications and variations are intended to fall within the scope of the present invention as defined by the appended claims and are intended to be granted the same protection when interpreted in accordance with the fair, lawful and equitable breadth. [Explanation of Symbols]

[0081] 1. Planar heating element Two-sided heating element 3-sided heating element 10 Base 20 Electrical conductors 20a wavy filament 30 Jordan curve 31 curve 32 curve 33 curve 34 curve 40 First Electrical Conductor 50 Second Electrical Conductor 51 Power supply point G m rectangular area

Claims

1. A planar heating element, A transparent substrate, The electrical conductor on the substrate and Includes, The aforementioned electrical conductor has a network structure, The network structure has the shape of a two-dimensional array of Jordan curves that are congruent with respect to translation, wherein any two adjacent Jordan curves are linked to each other in a first direction or a second direction, and the first and second directions are orthogonal to each other. Each of the Jordan curves is composed of only four curves that neither overlap nor intersect with each other, wherein one of the four curves is a finite-length reference curve with endpoints, one of the remaining three curves is obtained by the reflection of the reference curve in the first direction, one of the remaining two curves is obtained by the reflection of the reference curve in the second direction, and the remaining one curve is obtained by the inversion of the reference curve. The aforementioned reference curve is represented as the graph of a strictly monotonically increasing continuous function f(x): [a, b] → R, defined on the closed interval [a, b], which is a subset of the set R of all real numbers, where, 1) The equation 3(f(b)-f(a)) / 2 ≤ b-a ≤ 6(f(b)-f(a)) holds true. 2) The function f(x) is first differentiable on the open interval (a, b), ３） [Number 36] The following holds true, where f'(x) is the first derivative of the function f(x), 4) The graph of the function f(x) has one inflection point (x poi , f(x poi )) has, however a < x poi <b, ５） [Number 37] This is established, ６） [Number 38] This is true. Planar heating element.

2. In the planar heating element according to claim 1, (2a + b) / 3 ≤ x poi The equation ≤ (a + 2b) / 3 holds true. A planar heating element characterized by the following features.

3. In the planar heating element according to claim 1, (2f(a)+f(b)) / 3≦f(x poi The equation )≦(f(a)+2f(b)) / 3 holds true. A planar heating element characterized by the following features.

4. In the planar heating element according to claim 1, f'(x poi The condition )≦tan(π / 3) holds true. A planar heating element characterized by the following features.

5. In the planar heating element according to claim 1, The aforementioned reference curve is a part of a sigmoid curve, or a higher-order spline curve, or a B-spline curve. A planar heating element characterized by the following features.

6. In the planar heating element according to claim 1, The aforementioned reference curve is point-symmetric. A planar heating element characterized by the following features.

7. In the planar heating element according to claim 1, The above function f(x) is, [Number 39] or [Number 40] It is represented A planar heating element characterized by the following features.

8. In the planar heating element according to claim 1, The x-axis direction of the function f(x) is parallel to the second direction. A voltage is applied to the electrical conductor in the second direction. A planar heating element characterized by the following features.

9. A planar heating element, A transparent substrate, The electrical conductor on the substrate and Includes, The electrical conductor has a network structure divided into M rectangular regions in a first direction, where M is a predetermined odd number of 3 or more. The M rectangular regions are arranged in the first direction in an order according to the order relation of the elements of the set {x∈N: 1≦x≦M}, where N is the set of all positive integers. For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, a part of the network structure included in the m-th rectangular region among the M rectangular regions is a Jordan curve JC that is congruent to each other with respect to translation. m has a shape of a one-dimensional array or a two-dimensional array, provided that in the case of a two-dimensional array, any two adjacent Jordan curves JC m among the Jordan curves JC m are linked to each other in the first direction or the second direction, the second direction is orthogonal to the first direction, and in the case of a one-dimensional array, any two adjacent Jordan curves JC m among the Jordan curves JC m are linked to each other in the second direction. For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the Jordan curve JC m Each of these is composed of only four curves that neither overlap nor intersect with each other, except that one of the four curves is a finite-length reference curve RC with two endpoints. m And one of the remaining three curves is the reference curve RC in the first direction. m The curve obtained by the reflection of the other two curves is the reference curve RC in the second direction. m The curve obtained by reflection is the same as the reference curve RC. m This is a curve obtained by inverting the curve, For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the reference curve RC m f is a strictly monotonically increasing continuous function defined on the closed interval [a, b], which is a subset of the set R of all real numbers. m (x): [a, b] → R is represented as a graph, where, 1) 3(f m (b) - f m (a)) / 2≦b−a≦6(f m (b) - f m (a)) is true, 2) The function f m (x) is first differentiable on the open interval (a, b), ３） [Number 41] The following holds true, however, f m '(x) is the function f m The first derivative of (x) is, 4) The function f m The graph of (x) has one inflection point (x poi , f m (x poi )) has, however a < x poi <b, ５） [Number 42] This is established, ６） [Number 43] This is established, For (p, q) ∈ {x ∈ N: 1 ≤ x ≤ M} × {x ∈ N: 1 ≤ x ≤ M} that satisfy p ≠ q and p + q = M + 1, f p (b)-f p (a)=f q (b)-f q (a) This is established, For any r, s ∈ {x ∈ N: 1 ≤ x ≤ (M + 1) / 2}, r<s⇒f r (b)-f r (a)>f s (b)-f s (a) This is true. Planar heating element.

10. A planar heating element, A transparent substrate, The first electrical conductor on the substrate, Two second electrical conductors, each having one or two predetermined power supply points, Includes, The first electrical conductor has a network structure divided into M rectangular regions in a first direction, where M is a predetermined integer of 3 or more. The M rectangular regions are arranged in the first direction in an order according to the order relation of the elements of the set {x∈N: 1≦x≦M}, where N is the set of all positive integers. For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, a portion of the network structure contained within the m-th rectangular region out of the M rectangular regions is a Jordan curve JC that is congruent to each other with respect to translation. m It has the shape of a one-dimensional or two-dimensional array, however, in the case of a two-dimensional array, the Jordan curve JC m Any two adjacent Jordan curves JC m The elements are linked to each other in the first or second direction, the second direction being orthogonal to the first direction, and in the case of a one-dimensional array, the Jordan curve JC m Any two adjacent Jordan curves JC m They are linked to each other in the direction of the second Element, For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the Jordan curve JC m Each of these is composed of only four curves that neither overlap nor intersect with each other, except that one of the four curves is a finite-length reference curve RC with two endpoints. m And one of the remaining three curves is the reference curve RC in the first direction. m The curve obtained by the reflection of the other two curves is the reference curve RC in the second direction. m The curve obtained by reflection is the same as the reference curve RC. m This is a curve obtained by inverting the curve, For each m ∈ {x ∈ N: 1 ≤ x ≤ M}, the reference curve RC m f is a strictly monotonically increasing continuous function defined on the closed interval [a, b], which is a subset of the set R of all real numbers. m (x): [a, b] → R is represented as a graph, where, 1) 3(f m (b) - f m (a)) / 2≦b−a≦6(f m (b) - f m (a)) is true, 2) The function f m (x) is first differentiable on the open interval (a, b), ３） [Number 44] The following holds true, however, f m '(x) is the function f m The first derivative of (x) is, 4) The function f m The graph of (x) has one inflection point (x poi , f m (x poi )) has, however a < x poi <b, ５） [Number 45] This is established, ６） [Number 46] This is established, (i) When one of the two second electrical conductors has a predetermined power supply point: The power supply point of one of the two second electrical conductors and the power supply point of the other second electrical conductor are located on a straight line parallel to the second direction and passing through the K-th rectangular region of the M rectangular regions. (i-1) If 1 < K < M, For any r, s ∈ {x ∈ N: 1 ≤ x ≤ K}, r<s⇒f r (b)-f r (a)<f s (b)-f s (a) This is established, For any r, s ∈ {x ∈ N: K ≤ x ≤ M}, r<s⇒f r (b)-f r (a)>f s (b)-f s (a) This is established, (i-2) When K = 1, For any r, s ∈ {x ∈ N: 1 ≤ x ≤ M}, r<s⇒f r (b)-f r (a)>f s (b)-f s (a) This is established, (ii) When two of the power supply points are predetermined in each of the two second electrical conductors: Two of the two second electrical conductors are located at both ends of one of the second electrical conductors, and two of the other second electrical conductor is located at both ends of the other second electrical conductor. The aforementioned M is an odd number, For any r, s ∈ {x ∈ N: 1 ≤ x ≤ (M + 1) / 2}, r<s⇒f r (b)-f r (a)>f s (b)-f s (a) This is established, For any r, s ∈ {x ∈ N: (M+1) / 2 ≤ x ≤ M}, r<s⇒f r (b)-f r (a)<f s (b)-f s (a) This is true. Planar heating element.

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